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At least 235 records · Page 13

All sources NASA NTRS DOE OSTI

DOE OSTI2026

Deep Neural Networks are Adaptive to Function Regularity and Data Distribution in Approximation and Estimation

Deep learning has exhibited remarkable results across diverse areas. To understand its success, substantial research has been directed towards its theoretical foundations. Nev- ertheless, the majority of these studies examine how well deep neural networks can model functions with uniform regularities. In this paper, we explore a different angle: how deep neural networks can adapt to varying degrees of smoothness in functions and nonuni- form data distributions across different locations and scales. More precisely, we focus on a broad class of functions defined by nonlinear tree-based approximation methods. This class encompasses a range of function types, such as functions with uniform regularities and discontinuous functions. We develop nonparametric approximation and estimation theories for this class using deep ReLU networks. Our results show that deep neural networks are adaptive to the nonuniform smoothness of functions and nonuniform data distributions at different locations and scales. We apply our results to several function classes, and derive the corresponding approximation and generalization errors. The validity of our results is demonstrated through numerical experiments.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2001

Use of a genetic algorithm for the analysis of eye movements from the linear vestibulo-ocular reflex

It is common in vestibular and oculomotor testing to use a single-frequency (sine) or combination of frequencies [sum-of-sines (SOS)] stimulus for head or target motion. The resulting eye movements typically contain a smooth tracking component, which follows the stimulus, in which are interspersed rapid eye movements (saccades or fast phases). The parameters of the smooth tracking--the amplitude and phase of each component frequency--are of interest; many methods have been devised that attempt to identify and remove the fast eye movements from the smooth. We describe a new approach to this problem, tailored to both single-frequency and sum-of-sines stimulation of the human linear vestibulo-ocular reflex. An approximate derivative is used to identify fast movements, which are then omitted from further analysis. The remaining points form a series of smooth tracking segments. A genetic algorithm is used to fit these segments together to form a smooth (but disconnected) wave form, by iteratively removing biases due to the missing fast phases. A genetic algorithm is an iterative optimization procedure; it provides a basis for extending this approach to more complex stimulus-response situations. In the SOS case, the genetic algorithm estimates the amplitude and phase values of the component frequencies as well as removing biases.

Non-NASA Center↗

DOE OSTI2020

Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness

We report the accuracy of deep learning, i.e., deep neural networks, can be characterized by dividing the total error into three main types: approximation error, optimization error, and generalization error. Whereas there are some satisfactory answers to the problems of approximation and optimization, much less is known about the theory of generalization. Most existing theoretical works for generalization fail to explain the performance of neural networks in practice. To derive a meaningful bound, we study the generalization error of neural networks for classification problems in terms of data distribution and neural network smoothness. We introduce the cover complexity (CC) to measure the difficulty of learning a data set and the inverse of the modulus of continuity to quantify neural network smoothness. A quantitative bound for expected accuracy/error is derived by considering both the CC and neural network smoothness. Although most of the analysis is general and not specific to neural networks, we validate our theoretical assumptions and results numerically for neural networks by several data sets of images. The numerical results confirm that the expected error of trained networks scaled with the square root of the number of classes has a linear relationship with respect to the CC. We also observe a clear consistency between test loss and neural network smoothness during the training process. In addition, we demonstrate empirically that the neural network smoothness decreases when the network size increases whereas the smoothness is insensitive to training dataset size.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1994

The 1992/93 eclipse of 31 Cygni

Extensive new ultraviolet and optical spectra of an atmospheric eclipse define the physical properties throughout the wind and chromosphere of 31 Cyg. These data require mass loss of approximately 3 x 10(exp -8) solar masses/yr in a wind that may merge smoothly into the chromosphere. Considerations of how energy is injected into the wind, however, suggest that the chromosphere and wind are separate structures. Most, if not all, of the velocity structure in metallic lines, which we have heretofore simulated with Doppler widths in the range 15-25 km/s, results from differential expansion of the atmosphere. Electron densities in the inner R(sub star) of the chromosphere are in the range 1.5 x 10(exp 9) - 2 x 10(exp 8)/cu cm, which implies clumping of the gas. The ionization of oxygen and nitrogen is consistent with clumping by factors of 3-30 in the outer chromosphere, roughly the amount required to give enough gas pressure to support the chromosphere. Chromospheric gas in 31 Cyg becomes hotter with increasing height, thus with decreasing optical depth, in a way that seems similar for all the zeta Aur binaries. Excitation temperature for Fe II in 1992/93 rises from about 5000 K at the deepest points sampled to about 12,500 K high in the wind. Strengths of violet Balmer lines give an excitation temperature for hydrogen of 6200-6500 K above a radial mass column density of approximately 0.01 g/sq cm. This amount of excitation implies that Lyman-alpha is thermalized beneath about 4 x 10(exp -3) g/sq cm. The outer atmosphere was symmetrical to within a factor of 2 in 1992/93, although it was clearly variable at this level, and it had similar mass column densities as in 1982. One manifestation of the variability was a flow toward the B star at phases 0.013-0.022 spanning velocities 45-100 km/s. Several lines of evidence point to a complicated and variable ionization in the wind: At large distance from the K star, measured mass column densities are less by up to a factor of 3 than required by a smoothly flowing wind. Also, much of the gas beyond r = 350 solar radii = 1.75 R(sub star) has a negative radial velocity. Radial velocities of the shell lines imply the outer atmosphere rotates in the direction of orbital motion, possibly through the deflection of wind flow lines in this direction.

Eaton, Joel A.↗

NASA NTRS1972

Smoothing of functions of range and range rate measurements from earth orbiting satellites

It is shown that for satellites in circular Earth orbits with altitudes of 500 kilometers to 1500 kilometers, and for satellites in elliptical orbits with an approximate 4000 kilometer height of perigee, a high degree least squares polynomial (e.g., a 9th or 10th degree in some cases) is required to smooth both range and range rate data for purposes of input to orbit determination programs. In order to circumvent this problem, functions of range and range rate are smoothed with lower degree least squares polynomials (e.g., 3rd and 4th degree) and it is shown that under the above geometric constraints the standard deviation of fit can be reduced to levels commensurate with typical S-band tracking system resolution which is 1 to 2 meters in range and 0.005 meters/second in range rate for a 1 per second data rate. Also shown are the effects of Gaussian random noise, biases, and periodic noise. This analysis includes numerous examples applied to the 44 point data smoothing interval currently used in much of the operational preprocessing at the Goddard Space Flight Center.

Grenchik, T. J.↗

DOE OSTI2023

Cosmological constraints from the Subaru Hyper Suprime-Cam year 1 shear catalogue lensing convergence probability distribution function

Here we utilize the probability distribution function (PDF) of normalized convergence maps reconstructed from the Subaru Hyper Suprime-Cam (HSC) year 1 shear catalogue, in combination with the power spectrum, to measure the matter clustering amplitude S 8 = σ 8 $\sqrt {Ω_{m}/0.3}$. The large-scale structure’s statistical properties are incompletely described by the traditional two-point statistics, motivating our investigation of the PDF—a complementary higher-order statistic. By defining the PDF over the standard-deviation-normalized convergence map, we are able to isolate the non-Gaussian information. We use tailored simulations to compress the data vector and construct a likelihood approximation. We mitigate the impact of survey and astrophysical systematics with cuts on smoothing scales, redshift bins, and data vectors. We find S 8 = $0.860^{+0.066}_{–0.109}$ from the PDF alone and S 8 = 0.798$^{+0.029}_{–0.042}$ from the combination of the PDF and power spectrum (68% confidential level (CL)). The PDF improves the power-spectrum-only constraint by about 10%.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1981

A 5.57 hour modulation of the X-ray flux from 4U 1822-37

The X-ray flux of 4U 1822-37 is shown to be modulated with the 5.57 hr period of its optical counterpart. The X-ray light curve is two component, with a smooth sinusoidal-like 25% semiamplitude modulation and a 30 minute dip of approximately 0.2 in phase following the other minimum; the dip center occurs 0.04 in phase after the optical minimum. The X-ray spectrum is a relatively flat power law up to 17 keV, above which it steepens; iron emission is detected at 6.7 keV with a 4 keV FWHM and an equivalent width of 1100 eV. An excess below 2 keV is consistent with either a 0.25 keV thermal component of 350 eV equivalent width iron L emission. Modeling the eclipse for a 0.5-0.7 solar radius companion gives a system inclination of 70-79 deg and a spherical cloud radius of 0.2-0.3 solar radius. Models for the long-term modulation are considered, and a comparison of the properties of 4U 1822-37 with those of Cyg X-3 shows that they are similar systems.

White, N. E.↗

NASA NTRS1984

On long-term periodicities in the sunspot record

Sunspot records are systematically maintained, with the knowledge that an 11 year average period exists since about 1850. Thus, the sunspot record of highest quality and considered to be the most reliable is that of cycle eight through the present. On the basis of cycles 8 through 20, various combinations of sine curves were used to approximate the observed R sub MAX values (where R sub MAX is the smoothed sunspot number at cycle maximum). It is found that a three component sinusoidal function, having an 11 cycle and a 2 cycle variation on a 90 cycle periodicity, yields computed R sub MAX values which fit, reasonably well, observed R sub MAX values for the modern sunspot cycles. Extrapolation of the empirical functions forward in time allows for the projection of values of R sub MAX for cycles 21 and 22. For cycle 21, the function projects a value of 157.3, very close to the actually observed value of 164.5. For cycle 22, the function projects a value of about 107. Linear regressions applied to cycle 22 indicate a long-period cycle (cycle duration 132 months). An extensive bibliography on techniques used to estimate the time dependent behavior of sunspot cycles is provided.

Wilson, R. M.↗

NASA NTRS1986

An efficient method for solving the steady Euler equations

An efficient numerical procedure for solving a set of nonlinear partial differential equations, the steady Euler equations, using Newton's linearization procedure is presented. A theorem indicating quadratic convergence for the case of differential equations is demonstrated. A condition for the domain of quadratic convergence Omega(2) is obtained which indicates that whether an approximation lies in Omega(2) depends on the rate of change and the smoothness of the flow vectors, and hence is problem-dependent. The choice of spatial differencing, of particular importance for the present method, is discussed. The treatment of boundary conditions is addressed, and the system of equations resulting from the foregoing analysis is summarized and solution strategies are discussed. The convergence of calculated solutions is demonstrated by comparing them with exact solutions to one and two-dimensional problems.

Liou, M.-S.↗

NASA NTRS1992

Galaxies and gas in a cold dark matter universe

We use a combined gravity/hydrodynamics code to simulate the formation of structure in a random 22 Mpc cube of a cold dark matter universe. Adiabatic compression and shocks heat much of the gas to temperatures of 10 exp 6 - 10 exp 7 K, but a fraction of the gas cools radiatively to about 10 exp 4 K and condenses into discrete, highly overdense lumps. We identify these lumps with galaxies. The high-mass end of their baryonic mass function fits the form of the observed galaxy luminosity function. They retain independent identities after their dark halos merge, so gravitational clustering produces groups of galaxies embedded in relatively smooth envelopes of hot gas and dark matter. The galaxy correlation function is approximately an r exp -2.1 power law from separations of 35 kpc to 7 Mpc. Galaxy fluctuations are biased relative to dark matter fluctuations by a factor b about 1.5. We find no significant 'velocity bias' between galaxies and dark matter particles. However, virial analysis of the simulation's richest group leads to an estimated Omega of about 0.3, even though the simulation adopts Omega = 1.

Katz, Neal↗

NASA NTRS1941

Two-dimensional potential flow past a smooth wall with partly constant curvature

The speed of a two-dimensional flow potential flow past a smooth wall, which evinces a finite curvature jump at a certain point and approximates to two arcs in the surrounding area, has a vertical tangent of inflection in the critical point as a function of the arc length of the boundary curve. This report looks at a general theorem of the local character of the conformal function at the critical point as well as the case of the finite curvature jump.

Koppenfels, Werner Von↗

NASA NTRS1994

Nucleation, growth, and strain relaxation of lattice-mismatched 3-5 semiconductor epitaxial layers

We have investigated the early stages of evolution of highly strained 2-D InAs layers and 3-D InAs islands grown by metal-organic chemical vapor deposition (MOCVD) on (100) and (111)B GaAs substrates. The InAs epilayer/GaAs substrate combination has been chosen because the lattice-mismatch is severe (approximately 7.2 percent), yet these materials are otherwise very similar. By examining InAs-on-GaAs composites instead of the more common In(x)Ga(1-x)As alloy we remove an additional degree of freedom (x) and thereby simplify data interpretation. A matrix of experiments is described in which the MOCVD growth parameters - susceptor temperature, Thin flux, and AsH3 flux - have been varied over a wide range. Scanning electron microscopy, atomic force microscopy, transmission electron microscopy, and electron microprobe analysis have been employed to observe the thin film surface morphology. In the case of 3-D growth, we have extracted activation energies and power-dependent exponents that characterize the nucleation process. As a consequence, optimized growth conditions have been identified for depositing approximately 250 A thick (100) and (111)B oriented InAs layers with relatively smooth surfaces. Together with preliminary data on the strain relaxation of these layers, the above results on the evolution of thin InAs films indicate that the (111)B orientation is particularly promising for yielding lattice-mismatched films that are fully relaxed with only misfit dislocations at the epilayer/substrate interface.

Welser, R. E.↗

NASA NTRS1994

The structure of mass-loading shocks. 1: Comets

A new multifluid model to describe the solar wind-cometary ion plasma in the outer cometary coma is derived. This model is distinguished from previous multifluid models in that wave-particle effects are included explicitly. By considering hydrodynamic timescales, self-consistent dissipative terms (which correspond to the spatial diffusion of cometary ions) are derived, thereby allowing the structure of a cometary bow shock to be resolved. In the hypersonic limit, our diffusive multifluid model reveals an attractive connection to the one-fluid model formulated by Biermann et al. (1967), a connection which serves as an important guide when investigating our more complicated nonhypersonic diffusive model. The detailed model consists of cometary ions diffusing in a warm solar wind which is heated by the dissipation of Alfvenic turbulence (in situ and generated by the initial ring beam distribution). The quasi-parallel cometary shock is found to be at least an order of magnitude thicker than a quasi-perpendicular cometary shock. It is furthermore found that for undisturbed solar wind (sonic) Mach numbers in excess of approximately 6, the cometary bow shock (for both perpendicular and parallel cases) is smoothed completely by the cometary ions and no 'proton' or thermal subshock is necessary. Finally, a detailed comparison of the model with the plasma observations made at Halley by Giotto is undertaken. It is found that the observed shock structure, thickness, location, and plasma parameters compare well with the predictions of the theoretical model, particularly in the case of the quasi-parallel shock.

Zank, G. P.↗

NASA NTRS1996

Guidance of Nonlinear Systems

The paper describes a method for guiding a dynamic system through a given set of points. The paradigm is a fully automatic aircraft subject to air traffic control (ATC). The ATC provides a sequence of way points through which the aircraft trajectory must pass. The way points typically specify time, position, and velocity. The guidance problem is to synthesize a system state trajectory which satisfies both the ATC and aircraft constraints. Complications arise because the controlled process is multi-dimensional, multi-axis, nonlinear, highly coupled, and the state space is not flat. In addition, there is a multitude of possible operating modes, which may number in the hundreds. Each such mode defines a distinct state space model of the process by specifying the state space coordination, the partition of the controls into active controls and configuration controls, and the output map. Furthermore, mode transitions must be smooth. The guidance algorithm is based on the inversion of the pure feedback approximations, which is followed by iterative corrections for the effects of zero dynamics. The paper describes the structure and modules of the algorithm, and the performance is illustrated by several example aircraft maneuvers.

Meyer, George↗

NASA NTRS1997

Guidance of Nonlinear Systems

The paper describes a method for guiding a dynamic system through a given set of points. The paradigm is a fully automatic aircraft subject to air traffic control (ATC). The ATC provides a sequence of way points through which the aircraft trajectory must pass. The way points typically specify time, position, and velocity. The guidance problem is to synthesize a system state trajectory which satisfies both the ATC and aircraft constraints. Complications arise because the controlled process is multi-dimensional, multi-axis, nonlinear, highly coupled, and the state space is not flat. In addition, there is a multitude of possible operating modes, which may number in the hundreds. Each such mode defines a distinct state space model of the process by specifying the state space coordinatization, the partition of the controls into active controls and configuration controls, and the output map. Furthermore, mode transitions must be smooth. The guidance algorithm is based on the inversion of the pure feedback approximations, which is followed by iterative corrections for the effects of zero dynamics. The paper describes the structure and modules of the algorithm, and the performance is illustrated by several example aircraft maneuvers.

Meyer, George↗

NASA NTRS2012

Uniform Foam Crush Testing for Multi-Mission Earth Entry Vehicle Impact Attenuation

Multi-Mission Earth Entry Vehicles (MMEEVs) are blunt-body vehicles designed with the purpose of transporting payloads from outer space to the surface of the Earth. To achieve high-reliability and minimum weight, MMEEVs avoid use of limited-reliability systems, such as parachutes and retro-rockets, instead using built-in impact attenuators to absorb energy remaining at impact to meet landing loads requirements. The Multi-Mission Systems Analysis for Planetary Entry (M-SAPE) parametric design tool is used to facilitate the design of MMEEVs and develop the trade space. Testing was conducted to characterize the material properties of several candidate impact foam attenuators to enhance M-SAPE analysis. In the current effort, four different Rohacell foams are tested at three different, uniform, strain rates (approximately 0.17, approximately 100, approximately 13,600%/s). The primary data analysis method uses a global data smoothing technique in the frequency domain to remove noise and system natural frequencies. The results from the data indicate that the filter and smoothing technique are successful in identifying the foam crush event and removing aberrations. The effect of strain rate increases with increasing foam density. The 71-WF-HT foam may support Mars Sample Return requirements. Several recommendations to improve the drop tower test technique are identified.

Patterson, Byron W.↗

NASA NTRS2006

Adjoint Sensitivity Computations for an Embedded-Boundary Cartesian Mesh Method and CAD Geometry

Cartesian-mesh methods are perhaps the most promising approach for addressing the issues of flow solution automation for aerodynamic design problems. In these methods, the discretization of the wetted surface is decoupled from that of the volume mesh. This not only enables fast and robust mesh generation for geometry of arbitrary complexity, but also facilitates access to geometry modeling and manipulation using parametric Computer-Aided Design (CAD) tools. Our goal is to combine the automation capabilities of Cartesian methods with an eficient computation of design sensitivities. We address this issue using the adjoint method, where the computational cost of the design sensitivities, or objective function gradients, is esseutially indepeudent of the number of design variables. In previous work, we presented an accurate and efficient algorithm for the solution of the adjoint Euler equations discretized on Cartesian meshes with embedded, cut-cell boundaries. Novel aspects of the algorithm included the computation of surface shape sensitivities for triangulations based on parametric-CAD models and the linearization of the coupling between the surface triangulation and the cut-cells. The objective of the present work is to extend our adjoint formulation to problems involving general shape changes. Central to this development is the computation of volume-mesh sensitivities to obtain a reliable approximation of the objective finction gradient. Motivated by the success of mesh-perturbation schemes commonly used in body-fitted unstructured formulations, we propose an approach based on a local linearization of a mesh-perturbation scheme similar to the spring analogy. This approach circumvents most of the difficulties that arise due to non-smooth changes in the cut-cell layer as the boundary shape evolves and provides a consistent approximation tot he exact gradient of the discretized abjective function. A detailed gradient accurace study is presented to verify our approach. Thereafter, we focus on a shape optimization problem for an Apollo-like reentry capsule. The optimization seeks to enhance the lift-to-drag ratio of the capsule by modifyjing the shape of its heat-shield in conjunction with a center-of-gravity (c.g.) offset. This multipoint and multi-objective optimization problem is used to demonstrate the overall effectiveness of the Cartesian adjoint method for addressing the issues of complex aerodynamic design. This abstract presents only a brief outline of the numerical method and results; full details will be given in the final paper.

Nemec, Marian↗

NASA NTRS1991

The convergence rate of approximate solutions for nonlinear scalar conservation laws

The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the approximate solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates.

Nessyahu, Haim↗